Research Problems in Function Theory — Problem 4.28

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $P\in\mathbb R[z]$ have degree $d\ge2$, with no assumption that all of its zeros are real. Put $H(z)=P(z)^2+P'(z)$, a polynomial of degree $2d$. Let $N_{\mathrm{nr}}(H)$ count its zeros in $\mathbb C\setminus\mathbb R$ with their algebraic multiplicities.

Statement recorded in the supplied source.

Must $H$ have non-real zeros? More strongly, is \[N_{\mathrm{nr}}\left(P^2+P'\right)\ge d-1\] always true? Include arbitrary real-coefficient $P$, rather than only the source's earlier special case where all zeros of $P$ are real. [S2]

Short English statement

Does the square of every nonlinear real polynomial plus its derivative have non-real zeros, with at least degree minus one counted with multiplicity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304028 (AMR-022-4028). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 4.28, its complete Update 4.28, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.